Semi-discretization Method Research Articles

Investigation on chatter stability of thin-walled parts milling considering the effects of fixture support force and tool position

The chatter stability of thin-walled parts milling has garnered widespread attention in recent years, leading to numerous important findings. However, the effects of tool position and the fixture support force applied to the workpiece on the chatter stability have yet to be adequately studied. Therefore, this paper aims to explore the mechanisms by which these two factors affect the chatter stability of thin-walled parts milling. First, based on the structural characteristics of the milling system, a three-degree-of-freedom dynamic model for thin-walled parts milling is developed. Subsequently, modal impact tests are conducted to obtain the dynamic parameters under different fixture support forces and tool positions, and the evolution of these dynamic parameters under different fixture support forces and tool positions is explored in depth. Next, the semi-discrete method is employed to solve the dynamic model, analyzing the effects of fixture support force and tool position on the chatter stability of thin-walled parts milling. Finally, milling experiments are carried out to validate the accuracy of the model. The experimental results indicate that the stability prediction results from the established dynamic model are in agreement with the experimental results, demonstrating its effectiveness in assessing the chatter stability of thin-walled parts milling.

Stability and Bifurcations in Time-Delay Systems: A Novel and Efficient Blend of Methods

The goal of this paper is to introduce a hybrid technique, combining two existing methods conveniently, to analyze the stability and bifurcations of equilibrium points and periodic orbits in delay-differential equations of retarded type. One method is called the frequency-domain approach, an analytical tool based on control theory allowing to detect and represent periodic solutions emerging from the Hopf bifurcations, via Fourier series and harmonic balances. The second one, the so-called semi-discretization method, provides a numerical scheme to approximate the monodromy operator of a periodic solution, thus permitting to establish the stability and bifurcations of limit cycles as well as analyzing the equilibrium points. It is shown that a proper combination of these methods provides a straightforward strategy to study both local and global bifurcations in time-delay systems. The usefulness of the proposed approach is shown by three examples, where the results are compared with those given by the software DDE-BIFTOOL.

An existence result for integro-differential degenerate sweeping process with convex sets.

In this paper, we study the well-posedness in the sense of existence and uniqueness of a solution of integrally perturbed degenerate sweeping processes, involving convex sets in Hilbert spaces. The degenerate sweeping process is perturbed by a sum of a singlevalued map satisfying a Lipschitz condition and an integral forcing term. The integral perturbation depends on two time-variables, by using a semi-discretization method. Unlike the previous works, the Cauchy’s criterion of the approximate solutions is obtained without any new Gronwall’s like inequality.

Domain preserving and strongly converging explicit scheme for the stochastic SIS epidemic model

In this article, we construct a numerical method for a stochastic version of the Susceptible–Infected–Susceptible (SIS) epidemic model, expressed by a suitable stochastic differential equation (SDE), by using the semi-discrete method to a suitable transformed process. We prove the strong convergence of the proposed method, with order 1, and examine its stability properties. Since SDEs generally lack analytical solutions, numerical techniques are commonly employed. Hence, the research will seek numerical solutions for existing stochastic models by constructing suitable numerical schemes and comparing them with other schemes. The objective is to achieve a qualitative and efficient approach to solving the equations. Additionally, for models that have not yet been proposed for stochastic modeling using SDEs, the research will formulate them appropriately, conduct theoretical analysis of the model properties, and subsequently solve the corresponding SDEs.

An artificial neural network based deep collocation method for the solution of transient linear and nonlinear partial differential equations

A combined deep machine learning (DML) and collocation based approach to solve the partial differential equations using artificial neural networks is proposed. The developed method is applied to solve problems governed by the Sine–Gordon equation (SGE), the scalar wave equation and elasto-dynamics. Two methods are studied: one is a space-time formulation and the other is a semi-discrete method based on an implicit Runge–Kutta (RK) time integration. The methodology is implemented using the Tensorflow framework and it is tested on several numerical examples. Based on the results, the relative normalized error was observed to be less than 5% in all cases.

A priori estimates of the unsteady incompressible thermomicropolar fluid equations and its numerical analysis based on penalty finite element method

In this paper, the unsteady incompressible thermomicropolar fluid (UITF) equations are considered. Theoretically, some a priori regularity conclusions are presented firstly, which seem to be not available in the literatures. Numerically, a penalty finite element method (PFEM) for the UITF equations is studied, the Euler semi-implicit temporal semi-discrete method of the penalty UITF equations is proposed, the stability and the L2-H1 error estimates of the temporal discrete solutions are proved. Finally, the stability and the L2-H1 error estimates for the finite element fully-discrete approximation of the penalty UITF equations are rigorously proved. The accuracy and efficiency of the fully-discrete PFEM are demonstrated by some numerical examples.

Determination of an unknown coefficient in the Korteweg–de Vries equation

Abstract In this paper, a space-time spectral method for solving an inverse problem in the Korteweg–de Vries equation is considered. Optimal order of convergence of the semi-discrete method is obtained in L 2 {L^{2}} -norm. The discrete schemes of the method are based on the modified Fourier pseudospectral method in spatial direction and the Legendre-tau method in temporal direction. The nonlinear term is computed via the fast Fourier transform and fast Legendre transform. The method is implemented by the explicit-implicit iterative method. Numerical results are given to show the accuracy and capability of this space-time spectral method.

Stability and Bifurcation Analysis in a Discrete Predator–Prey System of Leslie Type with Radio-Dependent Simplified Holling Type IV Functional Response

In this paper, we use a semi-discretization method to consider the predator–prey model of Leslie type with ratio-dependent simplified Holling type IV functional response. First, we discuss the existence and stability of the positive fixed point in total parameter space. Subsequently, through using the central manifold theorem and bifurcation theory, we obtain sufficient conditions for the flip bifurcation and Neimark–Sacker bifurcation of this system to occur. Finally, the numerical simulations illustrate the existence of Neimark–Sacker bifurcation and obtain some new dynamical phenomena of the system—the existence of a limit cycle. Corresponding biological meanings are also formulated.

Adaptation of the composite finite element framework for semilinear parabolic problems

In this article, we discuss one type of finite element method (FEM), known as the composite finite element method (CFE). Dimensionality reduction is the primary benefit of CFE as it helps to reduce the complexity of the domain space. The number of degrees of freedom is greater in standard FEM compared to CFE. We consider the semilinear parabolic problem in a 2D convex polygonal domain. The analysis of the semidiscrete method for the problem is carried out initially in the CFE framework. Here, the discretization is carried out only in space. Then, the fully discrete problem is taken into account, where both the spatial and time components get discretized. In the fully discrete case, the backward Euler method and the Crank-Nicolson method in the CFE framework are adapted for the semilinear problem. The properties of convergence are derived and the error estimates are examined. It is verified that the order of convergence is preserved. The results obtained from the numerical computations are also provided.

Dynamic stability and control of vortex induced oscillations of tension leg platform tethers

Dynamic stability and control of sway oscillations of Tension Leg Platform (TLP) tether under regular vortex shedding are investigated. The TLP is modelled as a single degree of freedom simply supported uniform beam under pretension, assuming transverse oscillation to take place only in its fundamental natural mode. The tether is subjected to parametric excitation due to the heave motion of the TLP. The incremental harmonic balance (IHB) method is employed to investigate the period-1 and period-2 responses of the TLP as a function of the detuning parameter. The same system is then investigated under time-delayed state feedback control. Suitable control parameters are selected from the analysis of the local stability of the system equilibrium. The stability of solutions is analyzed by the semidiscretization method with the modified Floquet theory. The stable and periodic responses obtained by the IHB method are verified by directly integrating the equation of motion using the fifth-order Runge-Kutta algorithm at discrete points. Finally, the efficacy of the control scheme has been investigated in the plane of response amplitude versus excitation amplitude. The results show that appropriate selections of control parameters can effectively reduce the response amplitude to a great extent with the improved measure of stability.

Optimization of Redundant Degrees of Freedom in Robotic Flat-End Milling Based on Dynamic Response

With the advantages of large working space, low cost and more flexibility, industrial robots have become an important carrier in intelligent manufacturing. Due to the low rigidity of robotic milling systems, cutting vibrations are inevitable and have a significant impact on surface quality and machining accuracy. To improve the machining performance of the robot, a posture optimization approach based on the dynamic response index is proposed, which combines posture-dependent dynamic characteristics with surface quality for robotic milling. First, modal tests are conducted at sampled points to estimate the posture-dependent dynamic parameters of the robotic milling system. The modal parameters at the unsampled points are further predicted using the inverse distance weighted method. By combining posture-independent modal parameters with calibrating the cutting forces, a dynamic model of a robotic milling system is established and solved with a semi-discretization method. A dynamic response index is then introduced, calculated based on the extraction of the vibration signal peaks. The optimization model is validated through milling experiments, demonstrating that optimizing redundant angles significantly enhances milling stability and quality.

A divergence-free hybrid finite volume / finite element scheme for the incompressible MHD equations based on compatible finite element spaces with a posteriori limiting

We present a novel semi-implicit hybrid finite volume/finite element (FV/FE) method for the equations of viscous and resistive incompressible magnetohydrodynamics (MHD). The scheme preserves the divergence-free property of the magnetic field exactly on the discrete level, is second order accurate in space, and is stable in the limit of vanishing viscosity and resistivity. In particular, the MHD system is first split into a fluid and a magnetic subsystem. The former is then addressed via a semi-implicit hybrid FV/FE method. The latter belongs to a class of generalized advection-diffusion problems, for which we propose a novel semi-implicit scheme based on compatible finite element spaces and interpolation-contraction operators. In the case of vanishing resistivity, the new algorithm avoids the solution of a linear system at each time step, thanks to the framework of physical degrees of freedom, which allow to express the exterior derivative as the incidence matrix of an appropriate refinement of the original mesh. Regarding the timestepping, a local fully-discrete Lax-Wendroff-type evolution is proposed as an alternative to the classical semi-discrete method of lines. In order to suppress spurious oscillations in the presence of discontinuities or steep gradients in the magnetic field, a new simple but efficient a posteriori limiting procedure based on a nonlinear artificial resistivity is introduced. The novel methodology is validated with several test cases in two and three space dimensions using unstructured simplex meshes.

Moving Frames: Difference and Differential-Difference Lagrangians

This paper develops moving frame theory for partial difference equations and for differential-difference equations with one continuous independent variable. In each case, the theory is applied to the invariant calculus of variations and the equivariant formulation of the conservation laws arising from Noether's theorem. The differential-difference theory is not merely an amalgam of the differential and difference theories, but has additional features that reflect the need for the group action to preserve the prolongation structure. Projectable moving frames are introduced; these cause the invariant derivative operator to commute with shifts in the discrete variables. Examples include a Toda-type equation and a method of lines semi-discretization of the nonlinear Schrödinger equation.

A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

In this paper, a new efficient, and at the same time, very simple and general class of thermodynamically compatible finite volume schemes is introduced for the discretization of nonlinear, overdetermined, and thermodynamically compatible first-order hyperbolic systems. By construction, the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm. A very peculiar feature of our approach is that entropy is discretized directly, while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization. The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs, including both, conservative and non-conservative products, as well as potentially stiff algebraic relaxation source terms, provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law, such as the conservation of total energy density. The proposed family of finite volume schemes is based on the seminal work of Abgrall [1], where for the first time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented. We apply our new approach to three particular thermodynamically compatible systems: the equations of ideal magnetohydrodynamics (MHD) with thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning, the unified first-order hyperbolic model of continuum mechanics proposed by Godunov, Peshkov, and Romenski (GPR model) and the first-order hyperbolic model for turbulent shallow water flows of Gavrilyuk et al. In addition to formal mathematical proofs of the properties of our new finite volume schemes, we also present a large set of numerical results in order to show their potential, efficiency, and practical applicability.

Complicate dynamical properties of a discrete slow-fast predator-prey model with ratio-dependent functional response

Using a semidiscretization method, we derive in this paper a discrete slow-fast predator-prey system with ratio-dependent functional response. First of all, a detailed study for the local stability of fixed points of the system is obtained by invoking an important lemma. In addition, by utilizing the center manifold theorem and the bifurcation theory some sufficient conditions are obtained for the transcritical bifurcation and Neimark-Sacker bifurcation of this system to occur. Finally, with the use of Matlab software, numerical simulations are carried out to illustrate the corresponding theoretical results and reveal some new dynamics of the system. Our results clearly demonstrate that the system is very sensitive to its fast time scale parameter variable.

A filtered Chebyshev spectral method for conservation laws on network

We propose a spectral method based on the implementation of Chebyshev polynomials to study a model of conservation laws on network.We avoid the Gibbs phenomenon near shock discontinuities by implementing an exponential filter in the frequency space coupled with the Midpoint method for time marching. This strategy consists in adding a local viscosity into the model, and is suitable to contrast the spurious oscillations appearing in the profile of the solution. We prove the convergence of the semi-discrete method by using the compensated compactness theorem and exploiting the regularity of viscous profiles. Thanks to several simulations, we make a comparison between the implementation of the proposed method with a first order finite volume scheme.

Mixed Virtual Element approximation of linear acoustic wave equation

Abstract We design a Mixed Virtual Element Method for the approximated solution to the first-order form of the acoustic wave equation. In the absence of external loads, the semi-discrete method exactly conserves the system energy. To integrate in time the semi-discrete problem we consider a classical $\theta $-method scheme. We carry out the stability and convergence analysis in the energy norm for the semi-discrete problem showing an optimal rate of convergence with respect to the mesh size. We further study the property of energy conservation for the fully-discrete system. Finally, we present some verification tests as well as engineering applications of the method.

Numerical investigation of generalized tempered-type integrodifferential equations with respect to another function

This paper studies two efficient numerical methods for the generalized tempered integrodifferential equation with respect to another function. The proposed methods approximate the unknown solution through two phases. First, the backward Euler (BE) method and first-order interpolation quadrature rule are adopted to approximate the temporal derivative and generalized tempered integral term to construct a semi-discrete BE scheme. Second, the backward differentiation formula (BDF) and second-order interpolation quadrature rule are adopted to establish a semi-discrete second-order BDF (BDF2) scheme. Additionally, the stability and convergence of two semi-discrete methods are deduced in detail. To further demonstrate the effectiveness of proposed techniques, fully discrete BE and BDF2 finite difference schemes are formulated. Subsequently, the theoretical results of two fully discrete difference schemes are presented. Finally, the numerical results demonstrate the accuracy and competitiveness of the theoretical analysis.

A geometric approach to milling stability uncertainty

Deterministic solutions for the milling stability boundary are provided by frequency-domain, time-domain, and semi-discretization methods. While these deterministic solutions are valuable, it is essential to consider the uncertainty in the predicted stability boundary to enable optimum milling parameter selection. Prior efforts have implemented Type A (statistical) uncertainty evaluations. This paper provides a Type B (other means) analysis, where the stability uncertainty is represented by offset boundaries based on user-selected uncertainties in spindle speed and axial depth. The geometric approach is demonstrated for a selected milling system using a frequency-domain stability solution.

Dynamical integrity estimation in time delayed systems: A rapid iterative algorithm

The robustness of dynamical systems against external perturbations is crucial in engineering; however, it is often overlooked for the lack of methods for rapidly computing it. This paper proposes a novel algorithm for estimating the robustness of systems subject to time delay. More precisely, the algorithm iteratively estimates the so-called local integrity measure (LIM), that is, the radius of the largest hypersphere centered at the fixed point and located within its basin of attraction. Since time-delayed systems are infinite dimensional, initial conditions are restricted to a constrained type of initial functions. The semi-discretization method is used for rapidly simulating the dynamics of the systems, while trajectories are classified as converging or diverging using a subdivision of the reduced phase space into cells. The algorithm was tested on four different mechanical systems, and in all cases it very quickly provided an accurate estimation of the LIM. Moreover, it enabled the study of LIM trends in a multi-dimensional parameter space, which would have been unfeasible with alternative methods. This breakthrough in computational efficiency has important implications for engineering design, allowing for careful consideration of dynamical integrity and enhancing the safety and reliability of engineered systems, especially in the presence of time delays.